Genus of a pair of pants

Question

I think I might be confused about what a pair of pants is...but I see it as a 2 sphere with three holes. Or a disk with two disks removed. I don't understand why its genus is zero... Why not three? This surface has boundary, is there a special definition for the genus in such cases?What motivates it? Thanks in advance!

Answer 1

Yes, a sphere with three disks removed is a pair of pants. A disk with two disks removed is also a pair of pants. That is correct.

You might be confused because you may think the genus is the number of "holes" in a surface. But it's not the same kind of hole we are talking about. The hole in a torus isn't something that was removed from the surface. It's a feature of the topology. We "see" it when we view the torus in $\mathbb{R}^3$, but it's not really... there. The notion of "hole" isn't well-defined, so it's best to think about other definitions of the genus.

A first definition of the genus of a surface is the maximum number of non-self-intersecting closed curves that you can draw on a surface without disconnecting it. You cannot draw a closed curve on a pair of pants without disconnecting it (try to do it), so the genus is zero.

Another definition is more algebraic and works for orientable surfaces. Let $\chi$ be the Euler characteristic, $g$ be the genus, and $b$ the number of boundary components. Then $\chi = 2 - 2g - b$. The pair of pants clearly has $b = 3$. It is homotopy equivalent to a wedge of two circles, $S^1 \vee S^1$, so the Euler characteristic is $1-2=-1$. Plug that in, you get $-1 = 2-2g-3$ so $g = 0$ again.

Answer 2

Wikipedia gives a definition of genus for manifolds with boundary:

Alternatively, it can be defined in terms of the Euler characteristic $\chi$, via the relationship $\chi = 2 − 2g$ for closed surfaces, where $g$ is the genus. For surfaces with $b$ boundary components, the equation reads $\chi = 2 − 2g − b$.

So if you want to calculate the genus of a pair of pants, calculate the Euler characteristic first. This is a triangulation of something homeomorphic to a pair of pants:

I count 19 vertices, 40 edges and 20 triangles. So we have Euler characteristic $\chi = 19-40+20 = -1$. Since we have $b=3$ boundary components, we obtain $$-1 = 2-2g-3 \Leftrightarrow g=0$$